$$
\begin{aligned}
& \text { Given, }\left(1+x+x^2+x^3\right)^5=\sum_{k=0}^{15} a_k x^k \\
& \Rightarrow[(1+x)+x(1+x)]^5=\sum_{k=0}^{15} a_k x^k \\
& \Rightarrow(1+x)^{10}=a_0 x^0+a_1 x+a_2 x^2+\ldots+a_{15} x^{15} \\
& \Rightarrow \quad{ }^{10} C_0+{ }^{10} C_1 x+{ }^{10} C_2 x^2+\ldots+{ }^{10} C_{10} x^{10} \\
& \quad=a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots+a_{15} x^{15}
\end{aligned}
$$
On equating the coefficient of constant and even powers of $x$, we get
$$
\begin{aligned}
a_0 & ={ }^{10} C_0, a_2={ }^{10} C_2, \\
a_4 & ={ }^{10} C_4, \ldots, a_{10}={ }^{10} C_{10}, \\
a_{12} & =a_{14}=0 \\
\therefore \quad \sum_{k=0}^7 a_2 k & ={ }^{10} C_0+{ }^{10} C_2+{ }^{10} C_4+{ }^{10} C_6 \\
& +{ }^{10} C_8+{ }^{10} C_{10}+0+0 \\
& =2^{10-1}=2^9 \\
& =512
\end{aligned}
$$